Make the dumplings: Coarsely chop shrimp and ginger in a food processor. Transfer to a bowl. Stir in chives. Mound 1 teaspoon in center of each wrapper. Moisten edges with wet fingertips; fold over, pressing to seal. Cover with a damp paper towel; refrigerate until ready to use (up to 3 hours).
Make the soup: Bring stock, chile, and star anise to a simmer in a medium saucepan. Cook for 5 minutes. Add dumplings. Simmer until tender, 3 to 4 minutes.
Meanwhile, make the bok choy: Steam bok choy in water in a skillet over high heat, covered, until tender, about 3 minutes.
Stir mushrooms, soy sauce, radishes, and chives into soup. Serve soup with bok choy on the side.